789 research outputs found
A Stepwise Planned Approach to the Solution of Hilbert's Sixth Problem. II : Supmech and Quantum Systems
Supmech, which is noncommutative Hamiltonian mechanics \linebreak (NHM)
(developed in paper I) with two extra ingredients : positive observable valued
measures (PObVMs) [which serve to connect state-induced expectation values and
classical probabilities] and the `CC condition' [which stipulates that the sets
of observables and pure states be mutually separating] is proposed as a
universal mechanics potentially covering all physical phenomena. It facilitates
development of an autonomous formalism for quantum mechanics. Quantum systems,
defined algebraically as supmech Hamiltonian systems with non-supercommutative
system algebras, are shown to inevitably have Hilbert space based realizations
(so as to accommodate rigged Hilbert space based Dirac bra-ket formalism),
generally admitting commutative superselection rules. Traditional features of
quantum mechanics of finite particle systems appear naturally. A treatment of
localizability much simpler and more general than the traditional one is given.
Treating massive particles as localizable elementary quantum systems, the
Schrdinger wave functions with traditional Born interpretation appear
as natural objects for the description of their pure states and the
Schrdinger equation for them is obtained without ever using a
classical Hamiltonian or Lagrangian. A provisional set of axioms for the
supmech program is given.Comment: 55 pages; some modifications in text; improved treatment of
topological aspects and of Noether invariants; results unchange
The Role of Money in Supporting the Pareto Optimality of Competitive Equilibrium in Consumption-Loan Type Models
Perhaps the single most enduring theme in economics is that of the social desirability of the competitive mechanism. In its modern form, this theme occurs as the two basic theorems of welfare economics (see, in particular, Arrow). Our central concern in this paper is with the validity of the first of these two theorems—that every competitive equilibrium yields a Pareto optimal allocation—in idealized yet plausible models of intertemporal allocation in a market economy
Active Brownian Particles. From Individual to Collective Stochastic Dynamics
We review theoretical models of individual motility as well as collective
dynamics and pattern formation of active particles. We focus on simple models
of active dynamics with a particular emphasis on nonlinear and stochastic
dynamics of such self-propelled entities in the framework of statistical
mechanics. Examples of such active units in complex physico-chemical and
biological systems are chemically powered nano-rods, localized patterns in
reaction-diffusion system, motile cells or macroscopic animals. Based on the
description of individual motion of point-like active particles by stochastic
differential equations, we discuss different velocity-dependent friction
functions, the impact of various types of fluctuations and calculate
characteristic observables such as stationary velocity distributions or
diffusion coefficients. Finally, we consider not only the free and confined
individual active dynamics but also different types of interaction between
active particles. The resulting collective dynamical behavior of large
assemblies and aggregates of active units is discussed and an overview over
some recent results on spatiotemporal pattern formation in such systems is
given.Comment: 161 pages, Review, Eur Phys J Special-Topics, accepte
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